Preheat the oven to 400°. Season the chicken breasts with salt and black pepper. In a large ovenproof skillet, heat the olive oil. Add the chicken breasts, skin side down, and cook over moderately high heat until they are richly browned, about 3 minutes. Turn the chicken breasts and transfer the skillet to the oven. Roast for about 10 minutes, until just cooked through. Transfer the chicken to warmed plates.
Set the skillet over moderately high heat and add the shallots; cook until softened, about 3 minutes. Stir in the minced anchovies and crushed red pepper. Add the white wine and boil for 1 minute, scraping up any browned bits from the bottom of the pan. Add 1/3 cup of water and boil until the liquid is reduced to 3 table- spoons, about 1 minute. Remove the skillet from the heat and stir in the basil and lemon juice. Swirl in the butter and season the pan sauce with salt and pepper.
Pour the anchovy-basil pan sauce over the roasted chicken breasts, spooning the shallots all around, and serve right away.